Representation of heteroskedasticity in discrete choice models

The Multinomial Logit, discrete choice model of transport demand has several restrictions when compared with the more general Multinomial Probit model. The most famous of these are that unobservable components of utilities should be mutually independent and homoskedastic. Correlation can be accommodated to a certain extent by the Hierarchical Logit model, but the problem of heteroskedasticity has received less attention in the literature. We investigate the consequences of disregarding heteroskedasticity, and make some comparisons between models that can and those that cannot represent it. These comparisons, which use synthetic data with known characteristics, are made in terms of parameter recovery and estimates of response to policy changes. The Multinomial Logit, Hierarchical Logit, Single Element Nested Logit, Heteroskedastic Extreme Value Logit and Multinomial Probit models are tested using data that are consistent with various error structures; only the last three can represent heteroskedasticity explicitly. Two different kinds of heteroskedasticity are analysed: between options and between observations. The results show that in the first case, neither the Multinomial Logit nor the Single Element Nested Logit models can be used to estimate the response to policy changes accurately, but the Hierarchical Logit model performs surprisingly well. By contrast, in a certain case of discrete heteroskedasticity between observations, the simulation results show that in terms of response to policy variations the Multinomial Logit model performs as well as the theoretically correct Single Element Nested Logit and Multinomial Probit models. Furthermore, the Multinomial Logit Model recovered all parameters of the utility function accurately in this case. We conclude that the simpler members of the Logit family appear to be fairly robust with respect to some homoskedasticity violations, but that use of the more resource-intensive Multinomial Probit model is justified for handling the case of heteroskedasticity between options.